Definition
An objective function in linear programming is a mathematical expression that encapsulates the goal of a decision-making problem. It defines what needs to be maximized or minimized, such as profit, cost, or efficiency, based on specific variables and constraints. The objective function forms the basis for finding the optimal solution to the problem, as linear programming involves selecting the best possible outcome under given limitations.
For example, if a company wishes to maximize its profit (contribution) or minimize its production costs, the objective function would be formulated mathematically to represent these goals.
Examples
Example 1: Maximizing Profit
A company produces two products, A and B. The profit from Product A is $5 per unit, and the profit from Product B is $7 per unit. The objective function to maximize the profit (P) would be:
\[ P = 5A + 7B \]
Example 2: Minimizing Costs
A manufacturing firm wants to minimize its total cost. The cost to produce 1 unit of materials X and Y are $10 and $15, respectively. The objective function for minimizing the total cost (C) would be:
\[ C = 10X + 15Y \]
Frequently Asked Questions
What are some common types of objective functions?
- Maximization: Often used for profit, revenue, or efficiency.
- Minimization: Commonly applied to cost, waste, or time.
How is an objective function different from constraints?
- Objective Function: Defines the goal such as maximizing profits or minimizing costs.
- Constraints: Limits within which the objective function must operate, like resource limitations or budget caps.
Why is linear programming important in business?
Linear programming helps businesses optimize resource allocation, maximize profits, minimize costs, and make efficient decisions under constraints.
How do you determine the variables in an objective function?
Variables are determined based on the factors that directly impact the goal. For instance, in a profit-maximization problem, the variables would be the different products or services offered.
What tools can be used to solve linear programming problems?
Tools include software like Excel’s Solver, LINDO, MATLAB, and specialized programming languages like Python with libraries such as PuLP or SciPy.
Related Terms
- Linear Programming: A mathematical technique for optimization where a linear objective function is maximized or minimized subject to linear constraints.
- Constraints: Conditions or limitations imposed on the variables in an optimization problem.
- Feasible Region: The set of all possible points that satisfy the problem’s constraints.
- Optimization: The process of finding the best solution from all feasible solutions.
Online References
- Investopedia: Linear Programming
- Khan Academy: Linear Programming
- MIT OpenCourseWare: Optimization Methods
Suggested Books for Further Studies
- “Introduction to Operations Research” by Frederick S. Hillier and Gerald J. Lieberman
- “Operations Research: An Introduction” by Hamdy A. Taha
- “Linear Programming and Network Flows” by Mokhtar S. Bazaraa, John J. Jarvis, and Hanif D. Sherali
- “Convex Optimization” by Stephen Boyd and Lieven Vandenberghe
Accounting Basics: “Objective Function” Fundamentals Quiz
Thank you for exploring the fundamentals of the objective function in linear programming. Continue enhancing your understanding with further studies and practice.
